# AP Calculus BC : Chain Rule and Implicit Differentiation

## Example Questions

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### Find dy/dx by implicit differentiation:

Explanation:

To find dy/dx we must take the derivative of the given function implicitly. Notice the term  will require the use of the Product Rule, because it is a composition of two separate functions multiplied by each other. Every other term in the given function can be derived in a straight-forward manner, but this term tends to mess with many students. Remember to use the Product Rule:

Product Rule:

Now if we take the derivative of each component of the given problem statement:

Notice that anytime we take the derivative of a term with involved we place a "dx/dx" next to it, but this is equal to "1".

So this now becomes:

Now if we place all the terms with a "dy/dx" onto one side and factor out we can solved for it:

This is one of the answer choices.

### Find dx/dy by implicit differentiation:

Explanation:

To find dx/dy we must take the derivative of the given function implicitly. Notice the term  will require the use of the Product Rule, because it is a composition of two separate functions multiplied by each other. Every other term in the given function can be derived in a straight-forward manner, but this term tends to mess with many students. Remember to use the Product Rule:

Product Rule:

Now if we take the derivative of each component of the given problem statement:

Notice that anytime we take the derivative of a term with y involved we place a "dy/dy" next to it, but this is equal to "1".

So this now becomes:

Now if we place all the terms with a "dx/dy" onto one side and factor out we can solved for it:

This is one of the answer choices.

### Example Question #1 : Chain Rule And Implicit Differentiation

Use implicit differentiation to find the slope of the tangent line to  at the point .

Explanation:

We must take the derivative  because that will give us the slope. On the left side we'll get

, and on the right side we'll get .

We include the  on the left side because  is a function of , so its derivative is unknown (hence we are trying to solve for it!).

Now we can factor out a  on the left side to get

and divide by  in order to solve for .

Doing this gives you

.

We want to find the slope at , so we can sub in  for  and

.

### Example Question #2 : Chain Rule And Implicit Differentiation

Evaluate .

Explanation:

To find , substitute  and use the chain rule:

Plug in 3:

### Example Question #3 : Chain Rule And Implicit Differentiation

Evaluate .

Undefined

Explanation:

To find , substitute  and use the chain rule:

So

and

### Example Question #4 : Chain Rule And Implicit Differentiation

Evaluate .

Undefined

Explanation:

To find , substitute  and use the chain rule:

### Example Question #5 : Chain Rule And Implicit Differentiation

Evaluate .

Explanation:

To find , substitute  and use the chain rule:

So

and

### Example Question #6 : Chain Rule And Implicit Differentiation

Evaluate .

Explanation:

To find , substitute  and use the chain rule:

So

and

### Example Question #7 : Chain Rule And Implicit Differentiation

Explanation:

Consider this function a composition of two functions, f(g(x)). In this case, f(x) is ln(x) and g(x) is 3x - 7. The derivative of ln(x) is 1/x, and the derivative of 3x - 7 is 3. The derivative is then